Kernel theorems for the spaces of tempered ultradistributions

We give a simple proof of the Kernel theorem for the spaces of tempered ultradistributions of Beurling–Komatsu and of Roumieu–Komatsu types, by using the characterization of Fourier–Hermite coefficients of the elements of the spaces. As a consequence of the Kernel theorem, we have that the Weyl transform can be extended on spaces of tempered ultradistributions.